Functions f and g

Today in calculus class, we were finding the area between two curves. More specifically, as I pointed toward the graph on the board, I said, “Let’s find the area between functions f and g.” It sounded like, “f-n-g.” Or maybe that should be spelled like eff-in-gee.
The whole class started smiling and giggling. Once I realized what it sounded like, I had to smile too. Then I had to watch my speech and not connect f directly with g. They knew I was carefully choosing my words and got a kick out of everytime I needed to say something about “f and g”.
The thing is, I don’t know how many times I have said “f and g” in all my years of teaching. I don’t think I have said it much different in the past. Why hasn’t another class caught the similar sounds?
Well, shortly after that, we graphed the equations on an overhead graphing calculator. As I adjusted the window values to display just quadrant I on the calculator, I innocently said, “I like to show a little x-axis.” More smiles.
I have great students. God bless them all.

The Four Wives

Found this in my email inbox…
Once upon a time there was a rich King who had four wives. He loved the fourth wife the most and adorned her with rich robes and treated her to the finest of delicacies. He gave her nothing but the best.
He also loved the third wife very much and was always showing her off to neighboring kingdoms. However, he feared that one day she would leave him for another.
He also loved his second wife. She was his confident and was always kind,considerate and patient with him. Whenever the King faced a problem, he could confide in her, and she would help him get through the difficult times.
The King’s first wife was a very loyal partner and had made great contributions in maintaining his wealth and kingdom. However, he did not love the first wife. Although she loved him deeply, he hardly took notice of her!
One day, the King fell ill and he knew his time was short. He thought of his luxurious life and wondered, “I now have four wives with me, but when I die, I’ll be all alone.”
Thus, he asked the fourth wife, “I have loved you the most, endowed you with the finest clothing and showered great care over you. Now that I’m dying, will you follow me and keep me company?”
“No way!” replied the fourth wife, and she walked away without another word. Her answer cut like a sharp knife right into his heart.
The sad King then asked the third wife, “I have loved you all my life. Now that I’m dying, will you follow me and keep me company?”
“No!” replied the third wife. “Life is too good! When you die, I’m going to remarry!”
His heart sank and turned cold. He then asked the second wife, “I have always turned to you for help and you’ve always been there for me. When I die, will you follow me and keep me company?”
“I’m sorry, I can’t help you out this time!” replied the second wife. “At the very most, I can only walk with you to your grave.” Her answer struck him like a bolt of lightning, and the King was devastated.
Then a voice called out: “I’ll go. I’ll follow you no matter where you go.”
The King looked up, and there was his first wife. She was very skinny as she suffered from malnutrition and neglect. Greatly grieved, the King said, “I should have taken much better care of you when I had the chance.”
In truth, we all have four wives in our lives.
[Think about it for a moment before reading further.]

(more…)

May the road rise to meet you…

[Celtic Cross]

May the road rise to meet you;
May the wind be always at your back;
May the sun shine warm upon your face,
The rains fall soft upon your fields,
and until we meet again,
May God hold you in the palm of His Hand.

— an Irish blessing

He Died for Me

Hanging over the desk in the priest’s office is a picture of the crucifixion. The point of view is from above the Cross, kind of high above His right hand and a little out in front, looking downward toward Him and this vast crowd of people standing around the hill staring up at Jesus. You cannot see the ground in front of the Cross where I presume Mary, John, and others were standing. No, the emphasis was on the crowd of people standing in the shadow of the Cross, but there was no shadow. It was more of an illumination. Jesus on the Cross centered in the foreground, the people wrapped all around in the background, focused on Him.
The people in the crowd represent all parts of the world from all of history. There is a caveman, kneeling. There are Asians, Africans, Native Americans, Aborigines, Eskimos, and Europeans. There were people dressed in jeans and t-shirts, Victorian dress, medieval peasants, ancient Roman togas, and so on. Off to one side is a nun dressed in her black habit, standing next to a young woman in cutoffs and halter top, giving the possible appearance of a prostitute. Which woman is more atuned to her sexuality? In the middle to the right stands an astronaut in a space suit. The reflection in his visor is the Christmas Star over the small village of Bethlehem. There is a sense of peace in all of their expressions as they gaze at His death on the Cross, a gift meant for all people for all time.
Back in November, in a moment of prayerful meditation, I was trying to concentrate on the reading for the day, and the image of this picture kept coming to mind. It intersected with a memory of someone saying that Christ would have died on the Cross for me if I was the only person in the universe. Then my imagination took control—
There I was, in the middle of that picture, surrounded by other people looking up at Jesus on the Cross. I glanced down and noticed the dust of the desert on my shoes. I felt the pebbles and dirt shift ever so slightly as I shifted my weight to the other foot. I looked up into a sky of soft powdery blue with a hint of white puffy clouds low on the horizon. The mid-afternoon sun was warm on my face. All was silent except for the gentle rustle of cloths in the cool breeze.
I looked back up to focus on Jesus on the Cross. He was dead. The drama of the Passion that lead up to this moment was complete. I knew what was going to happen later in the afternoon, and especially on Sunday morning. I understood the source of the peace I saw in the faces of all the other people standing next to me. I felt the peace too, but not completely. I had a haunting deep sense of guilt weighing heavy on my heart. My sin was responsible for this man being on the Cross. He had to die because of me.
Slowly at first, then more quickly, the people in the crowd started to disappear. In a few moments, I knew I would be the only one remaining. It would soon be time to face this guilt inside my heart. I would have to face it alone with Him. Instead of becoming anxious, I felt a certain measure of peace. I wondered if I should be afraid. How many times did He say not to be? As the others in the crowd disappeared, I expected the weight of my guilt to grow within me. There is animity in a crowd, a sharing of responsiblity that falsely disseminates the guilt. No, the weight of my guilt did not change. I knew it was mine. There was no fooling myself. No fooling Him.
Then the last person disappearded. There I was, alone. Alone before the Cross with Jesus dead on it. I continued to look up at Him. The sun was still warm on my face. The gentle breeze continued to playfully tossing my hair across my forehead. Should I prostrate myself on the ground before Him? I stood still. The silence of infinity enveloped me.
I bowed my head slightly as if gazing into my heart, searching for the heavy brick of guilt hiddened in a corner. There it was. I looked back up at Jesus and said in my mind, “I’m sorry. So sorry.” That’s all I could say. No excuses. No tears. “I’m sorry for my sin. Please forgive me.”
Somewhere in the back of my mind, I remembered reading something about Jesus asking St. Jerome in a vision why he had not given Him everything. Jerome replied, “Lord, I have devoted my life to your service. I have given you all my works, all my love, all my praise, everything.” Jesus replied, “No, you haven’t given me your sins.”
In an inward gesture of reaching my hands upward, I said in mind, “Lord, take my heart. Take all of it. All my love, all my joys, all my sorrow, all my sins, all my guilt. It is all yours.”
As I dropped my head again, I softly whispered, “I am Yours.”
A moment later, I looked upward at the Cross again, gazing deeply at His face. The crown of thorns still pierced His lifeless flesh. The trickles and streams of blood were dried and crusted in His hair and across His face. The cuts were still open and the bruises were blue and black and swollen.
On one side of the threshold lies pain, sorrow, loss, guilt, and death. By letting go—surrendering—one steps into the threshold of transformation, through the Pashcal Mystery of the Cross, and emerges into happiness, joy, victory, freedom, and life.
I looked one more time up at His face on the Cross. It seemed almost like He was smiling. The heavy brick of guilt was gone from my heart. He had died for me, and I was glad to receive His gift. A gladnes that only comes from grace.
The Way stood in front of me now. Lord help me. I will follow.

Happy Pi Day!

Ten decimals are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimals would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful microscope. — Simon Newcomb

[Energizer Bunny]

It’s March 14, and any student of mathematics should recognize today as Pi Day. Mathematicians do not get many days in the year to celebrate math, but here is one.

π is defined as the ratio of the circumference of a circle and its diameter. π is called an irrational number because it cannot be written as a rational number, like fractions. The ancient Greeks thought fractions were rational and logical (most modern people, especially students beg to differ). The irrational numbers were considered illogical. As I tell my students, π is the Queen of the Irrationals because it is the most famous of the irrational numbers.

When irrational numbers are written in decimal form, their digits do not terminate and do not repeat with no patterns. Kinda crazy. (Rational numbers either terminate or repeat.) That means irrational numbers are the Energizer Bunny of decimal numbers—they keep on going and never end. The current record for the most known decimal places of π is 1.241 trillion digits!